HG 

.11-3 


UC-.'^PM, ....MIIIIliHIil 


$B    37   155 


LIBRARY 

OF  THE 

University  of  California. 

GIFT  OF 

Class 

THE    DEVELOPMENT 

OF 

INSURANCE    MATHEMATICS. 


TWO   LECTURES 

Delivered  before  the  Students  in  the  School  of  Commerce 
of  the  University  of  Wisconsin,  the  Fall  Term  of  1901. 


BT 

MILES  MENANDER  DAWSON,  Consulting  Actuary. 


FIRST   LECTURE, 
Preliminary    Development,    Decimal  System,   Series,    Logarithms,    Per- 
mutations and  CJombinations,  Probabilities. 

SECOND   LECTURE. 

Evolution   of   Actuarial   Science,    Annuities   Certain,   Mortality    Tables, 
Actuarial  Science  Proper. 


1901. 
THE  INSURANCE  PRESS, 

Nuw    YOBK. 


THE  DEVELOPMENT  OF  INSDRAHCE  MATHEMATICS. 


FIRST  LECTURE— PREIilMLN-ARY  DEVELOPMENT. 

It  is  most  interesting  to  study  the  growth  and  evolution  of  any 
branch  of  science  and  to  observe  how  it  sprang  from  some  other 
branch  of  science,  and  what  things  had  to  be  known  as  a  prepara- 
tion for  the  dawning  of  the  new  learning.  It  is  a  far  cry,  perhaps, 
from  actuarial  science  to  the  decimal  system  of  notation,  and  yet, 
had  it  not  been  for  the  invention  of  that  system  it  may  well  be 
doubted  whether  arithmetic  would  ever  have  reached  the  stage 
which  renders  mathematical  problems  that  would  otherwise  be 
difficult  of  operation,  simple  and  easy.  Addition,  which  is  now 
so  simple,  albeit  so  laborious  a  matter,  was  exceedingly  difficult 
under  the  old  systems  of  notation.  A  student  may  readily  form 
some  conception  of  the  difficulties  if  he  will  range  a  number  of 
Eoman  numerals  side  by  side,  or  one  under  the  other — one  ar- 
rangement is  about  as  convenient  as  the  other — and  proceed  to 
add.  Multiplication,  which  is  continued  addition,  was  even  harder. 
It  follows  that  operations  which  must  be  performed  upon  a  very 
large  scale,  in  order  that  actuaries  may  do  their  tasks,  were  then 
almost  impossible  upon  anything  but  a  limited  scale. 

Almost  equally  absurd  it  may  at  first  seem  to  declare  that  the 
new  science  could  not  have  come  into  existence  until  after  algebra 
was  discovered;  but  it  will  be  seen  to  be  perfectly  reasonable  to 
say  this  when  it  is  taken  into  account  that  actuarial  science  is 
merely  a  department  of  algebra,  and  that  all  its  operations  are 
algebraic.  The  form  which  all  its  formulas  take  is  that  of  equa- 
tions, and  they  are  evolved  from  other  equations  by  purely  alge- 
braic processes. 

Algebra  and  decimal  notation  were  both  introduced  into  Europe 
by  the  Arabs  through  the  Moorish  schools  of  Spain  and  Africa,  m 
the  thirteenth  century.  The  first  mathematical  books  to  be  stud- 
ied were  those  of  the  Arabian  authors  and  Arabian  translations  of 


iionqi 


4  DEVELOPMENT  OF 

Euclid,  Archimedes,  Appolonius  and  Ptolemy.  The  stndy  was  for 
a  long  time  condemned  by  the  church  authorities,  although  one 
of  the  popes  was  a  mathematician.  It  was  connected  in  the  minds 
of  the  people  with  astrology  and  alchemy  and,  indeed,  most  of 
the  early  mathematicians  were  astrologers  also. 

The  following  things  needed  to  develop  in  mathematics  before 
actuarial  science  could  begin  its  evolution:  Decimal  notation, 
decimal  fractions — desirable,  not  necessary — series  and  summa- 
tion, logarithms,  permutations  and  combinations,  probabilities. 

Decimal  fractions  were  brought  in  by  Pitiscus  in  1617.  Their 
use,  however,  did  not  become  common  until  after  1700  and,  indeed, 
so  little  was  known  for  a  long  time  of  the  discovery  of  Pitiscus  that 
the  honor  of  the  invention  was  claimed  by  mathematicians  of  a 
later  date. 

Series  were  treated  by  the  ancient  mathematicians,  but  in  a 
very  fragmentary  and  imperfect  manner,  algebra  being  unknown. 
Euclid  and  Appolonius,  among  others,  discussed  the  subject  some- 
what, but  it  was  never  given  the  attention  that  was  necessary  in 
order  to  make  the  development  of  annuities  possible  until  in  the 
seventeenth  and  eighteenth  centuries,  when  Leibnitz,  Montmort, 
James  (or  Jacob)  and  John  Bernoulli,  La  Place,  Gauss  and  Cauchy 
threshed  out  the  matter  thoroughly. 

Even  with  a  perfected  decimal  system,  both  of  integral  and 
fractional  notation,  and  in  spite  of  the  perfection  of  algebra  as  a 
mathematical  art,  it  would  have  been  most  difficult  to  perform  the 
operations  required  in  compound  discount  computations,  as  in  an- 
nuities certain,  without  the  aid  of  logarithms,  which  make  it  al- 
most as  easy  to  find  the  present  value  of  a  sum  due  in  1,000  years 
as  if  the  term  were  a  single  year.  In  no  branch  of  mathematical 
science,  perhaps,  has  the  invention  of  logarithms  proven  a  greater 
blessing.  Their  invention,  as  is  well  known,  is  due  to  Napier,  and 
dates  from  about  1610.  They  were  soon  afterward  improved  by 
Briggs,  who  also  suggested  the  decimal  base,  which  made  logar- 
ithms much  easier  to  employ  and  far  more  useful. 

Annuities  certain  grew  so  inevitably  out  of  the  formulas  for  the 
summation  of  geometric  series  that  it  is  impossible,  perhaps,  to 
mark  out  closely  the  period  when  the  first  mention  of  the  subject 
is  found.    Indeed,  although  the  topic  has  a  deal  of  interest  of  itself. 


INSURANCE  MATHEMATICS  5 

and  is  generally  considered  abstruse  and  difficult  by  students, 
problems  in  annuities  certain,  like  all  compound-interest  prob- 
lems, were  originally  merely  set  as  exercises  in  series,  and  no 
thought  was  given  that  such  might  be  the  foundation  for  a  new 
branch  of  the  science.  As  a  separate  branch  of  mathematical  re- 
search, we  do  not  find  that  annuities  certain  appear  before  the 
beginning  of  the  eighteenth  century. 

As  series  and,  perhaps,  logarithms,  were  necessary  first  steps  to 
annuities  certain,  which  is  one  of  the  three  legs  upon  which  actu- 
arial science  will  be  found  to  stand,  so  in  like  manner  permuta- 
tions and  combinations  had  to  precede  probabilities,  which  is  the 
second  of  the  three  supports  of  actuarial  science. 

Perhaps,  as  the  subject  first  appeared  in  mathematics,  nothing 
was  ever  more  unpromising  so  far  as  practical  results  were  con- 
cerned than  permutations  and  combinations.  That  it  could  ad- 
vantage anybody  to  know  how  many  different  arrangements  could 
be  made  of  n  things  taken  k  at  a  time,  seemed  too  ridiculous  to 
talk  about.  Often  has  it  proved,  however,  that  the  topic  which 
was  considered  mere  mathematical  sport  has  been  found  most 
useful  in  some  field  which  was  awaiting  that  very  discovery  to  un- 
lock new  treasures  of  practically  valuable  learning.  So  it  was  in 
the  case  of  the  apparently  idle  speculations  as  to  the  numl  er  of 
arrangements  of  different  things  which  one  could  make;  what 
seemed  to  the  schoolmen  only  elegant  and  frivolous  speculation, 
highly  amusing  because  of  the  frequently  startling  results,  was  tho 
missing  word  to  unseal  the  secrets  of  probabilities. 

The  subject  was  not  unknown  to  the  ancients,  either  of  Greece 
or  of  India.  Aristotle,  whose  death  took  place  322  B.  C,  suggestea 
the  solutions  of  such  problems,  and,  before  his  time,  Arjabhatta,, 
the  Indian  savant,  treated  certain  phases.  The  modern  develop- 
ment began  in  the  later  years  of  the  sixteenth  century,  in  the  re- 
searches of  Cardan,  which,  however,  were  not  printed  until  1663. 
He  was  followed  soon  by  Pascal,  Leibnitz  and  James  (or  Jacob) 
Bernoulli. 

Concurrently  with  the  development  of  the  theories  of  permu- 
tations and  combinations  came  their  application  in  the  science  of 
probabilities.  For  Cardan,  who  died  in  1576,  wrote  the  first  known 
essay  on  the  subject  of  probabilities,  though  the  same  was  not  pub- 


6  DEVELOPMENT  OF 

lished  until  1663.  It  was  known  by  the  title  "De  ludo  aleae" — 
^^Concerning  the  game  of  dice." 

In  ancient  times  they  evidently  had  the  conception  that  the 
value  of  a  chance  or  probability  could  be  valued;  for,  as  we  shall 
see,  the  courts  occasionally  set  values  upon  life  annuities.  More- 
over, loans  were  made  by  venturesome  capitalists  upon  ships  and 
cargoes,  repayable  at  more  than  the  usual  interest,  but  forgiven 
entirely  if  the  property  were  destroyed.  The  similarity  to  insur- 
ance is  clear,  and  it  is  also  evident  that  there  was  some  rough 
fashion  by  which  what  was  believed  to  be  the  value  of  the  hazard 
was  ascertained.  But  if,  as  frequently  in  our  days,  there  was 
empirical  guessing  at  the  value  of  hazards,  there  was  no  attempt 
made  to  calculate  these  values,  so  far  as  is  now  known. 

Galileo,  at  some  time  during  his  life,  which  closed  in  1642,  also 
wrote  briefly  upon  the  same  subject  as  Cardan. 

But  the  first  work  of  importance  consists  of  a  series  of  letters 
by  Pascal  in  reply  to  certain  inquiries  of  the  Chevalier  de  Mere — 
by  some  thought  to  have  been  mythical — concerning  the  chances 
of  play.  The  first  of  these  letters  was  written  in  1654,  and  they 
were  published  in  1679.  In  them  some  of  the  most  intricate 
problems  of  probabilities  have  been  worked  out.  The  philosopher 
was  not  satisfied  to  deal  merely  with  the  simple  questions  that 
lay  u-  -on  the  surface,  but  boldly  assailed  the  most  complicated.  In 
the  course  of  his  investigations  he  discovered  the  "arithmetical  tri- 
angle'^ and  "figurative  numbers,"  which  have  puzzled  the  drones 
;.mong  college  students  from  that  day  to  this,  but  have  served 
nany  good  purposes  in  mathematics,  and  stood  ready  to  welcome 
ihe  binomial  theorem  at  a  later  date.  Merely  to  mention  this  one 
discovery,  sot  forth  in  his  letters,  may  serve  to  indicate  that  he 
did  not  indulge  in  mere  child's  play  in  his  primer  of  probabilities. 
His  contemporary,  Fermat,  also  took  part  in  the  discussion. 

The  next  work  upon  the  subject  was  by  Christian  Hu5^gens. 
Like  the  first  writing  dealing  with  probabilities,  it  was  an  attempt 
to  compute  the  chances  with  dice.  Its  title  was  "De  Eatiociniis 
in  Ludo  Aleae."    The  pamphlet  appeared  in  1657. 

After  this  there  was  a  long  pause.  The  situation  in  mathemat- 
ics was  at  the  time  about  as  follows :  After  the  subject  had  been 
reintroduced  it  was  for  a  long  time  strenuously  opposed  by  church 


INSURANCE  MATHEMATICS  7 

influences.  Moreover,  it  was  by  the  schools  despised  as  forming  no 
part  of  a  liberal  education.  The  universities  for  a  long  time  knew 
it  not.  The  connection  of  mathematics  with  astrolog}' — all  who 
studied  the  stars  were  esteemed  astrologists  in  those  days — ^with 
alchemists — all  students  of  nature  were  considered  alchemists — 
did  not  add  to  its  repute.  The  church  was  suspicious  of  all 
learning  which  did  not  emanate  from  itself.  But  at  about  this 
time,  as  a  result  of  the  serious  attention  to  mathematics,  which 
some  of  the  greatest  minds  of  the  age  had  paid,  and  also,  perhaps, 
because  the  control  of  education  was  passing  into  secular  hands, 
mathematics  was  beginning  to  make  its  way  into  the  universities. 
But  this  was  only  in  the  more  abstract  forms.  Most  of  its  appli- 
cations to  chemistry,  astronomy  and  other  sciences  were  yet  to 
be  made.  The  least  promising  of  these  applications  seemed  to  be 
to  probabilities ;  for  not  only  was  it  denied  that  there  was  or  could 
be  such  a  science,  but  the  use  of  it,  unless  to  encourage  gaming, 
nobody  could  see.  Therefore  the  first  attempts  to  solve  problems 
of  chance  were  ridiculed  by  most  scholars,  who  denied  both  that 
it  could  be  done  and  that  it  would  be  of  any  use  if  it  were  done. 
Doubtless,  therefore,  the  science  would  have  been  a  long  while 
getting  a  start,  had  not  a  recluse  like  Pascal,  fond  of  all  specula- 
tions that  tested  his  skill  in  analysis  and  synthesis,  and  altogether 
careless  whether  his  speculations  were  useful  or  not,  been  tempted 
to  essay  the  task  of  estimating  chances. 

Notwithstanding  this  auspicious  beginning,  however,  there  was 
nothing  further  upon  the  subject  for  nearly  forty  years,  and  noth- 
ing of  much  consequence  for  more  than  fifty  years.  In  1693  the 
first  study  of  the  subject  appeared  in  English;  its  title  is  "An 
Arithmetical  Paradox  Concerning  the  Chances  of  Lotteries."  Thus 
but  little  over  two  hundred  years  ago  actuarial  science,  which  was 
first  brought  to  its  full  development  in  Great  Britain,  found  lodg- 
ment there. 

This  first  English  work  was  of  minor  importance.  It  merely 
treated  some  practical  phases  of  the  subject,  and  was  by  no  means 
thorough.  But  there  had  been  for  many  years  preparing  a  really 
great  work  upon  the  subject,  the  first  of  all  great  works.  "Ars 
Conjectandi,"  by  Jacob  (often  called  James)  Bernoulli,  was,  of 
course,  completed  before  the  author's  death  in  1705,  but  it  was 


8  DEVELOPMENT  OF 

not  brought  out  until  1713.  It  was  edited  and  published  by  the 
nephew  of  the  author,  Nicholas,  son  of  John  Bernoulli;  and  from 
this  date  the  subject  of  probabilities  afforded  this  family  opportu- 
nities to  distinguish  themselves  even  unto  the  third  generation. 

That  "Ars  Conjectandi"  marks  the  close  of  an  old  era  and  the 
beginning  of  a  new  is  sufficiently  attested  by  the  following:  It 
was  written  in  Latin,  a  language  until  then  practically  always 
employed  as  a  vehicle  for  learning,  and  it  dealt  with  the  newest 
application  of  what  was  itself  a  new  science,  algebra.  It  was  at 
the  turning  point.  Latin  was  going  out;  science  was  coming  in. 
The  book  was  not  a  mere  sketch  or  study.  Jacob  Bernoulli  had 
toiled  during  the  last  years  of  his  life  most  indefatigably  and 
without  paying  the  least  attention  to  the  circumstance  that  the 
whole  speculation  was  by  most  learned  men  deemed  vain  and  un- 
profitable. When  he  laid  down  his  pen,  disappointed  even  in  his 
hope  that  his  manuscript  should  be  published,  he  had  developed 
the  scraps  of  speculation  into  a  demonstrated  science.  And  al- 
though even  at  a  later  day  persons  could  be  found  who  were  so 
dull  as  not  to  apprehend  his  truths,  and  so  intensely  prejudiced  as 
to  decide  against  him  on  a  priori  and  most  fanciful  grounds,  he 
had  fixed  the  fundamental  lines  for  the  investigation  of  probabili- 
ties where  they  have  ever  since  been  found.  His  work  dealt  with 
all  the  simpler  problems  and  with  many  of  the  compound  problems, 
including  some  of  the  most  intricate.  It  was  long  the  storehouse 
from  which  later  mathematicians  drew  their  material,  and  what  is 
owing  to  Bernoulli  it  would  now  be  very  difficult  to  overestimate. 
He  found  probabilities  a  subject  for  occasional  and  wholly  tenta- 
tive reasoning;  he  left  it  a  science. 

After  the  death  of  Jacob  Bernoulli,  but  before  the  publication 
of  his  book,  there  appeared  in  France,  in  1708,  a  pamphlet  entitled 
"An  Essay  on  the  Analysis  of  the  Game  of  Hazard."  The  author 
was  the  great  Montmort.  The  work  has  value,  but  need  only  be 
here  mentioned  to  show  that  the  speculations  of  Pascal  had  not 
been  fruitless  in  his  own  country.  There  for  some  time  the  topic 
was,  however,  following  Pascal's  lead,  treated  only  as  to  the  proba- 
bilities in  various  games  of  chance. 

The  next  great  mathematician  to  take  up  the  matter  was  also 
a  Frenchman,  De  Moivre;  but  he  was  an  exile  in  London,  and 


INSURANCE  MATHEMATICS  9 

his  books  were  written  in  English.  His  fame  was  also  achieved 
in  England,  and  he  is  known  as  a  great  English  mathematician. 
I  have  said  that  his  books  were  written  in  English,  which  also  was 
the  case  as  to  all,  save  a  pamphlet  entitled  "De  Mensnra  Sortis," 
which  was  published  in  1711.  This  was  followed  by  his  great 
work,  ^T)octrine  of  Chances,"  which  was  published  in  1718,  and 
was  the  second  book  to  treat  the  entire  subject  of  probabilities, 
and  the  first  to  appear  in  a  modern  language.  This  book  re- 
mained the  standard  text -book  upon  the  subject  for  a  long  time, 
and  during  the  author's  lifetime  two  revised  editions,  the  last 
much  enlarged,  also  appeared. 

Meanwhile  the  Bernoullis  were  keeping  interest  in  the  subject 
alive  in  other  countries.  Nicholas  Bernoulli  even  preceded  the 
publication  of  his  uncle's  "Ars  Conjectandi"  by  his  o^vn  pamphlet, 
'•'Specimens  of  the  Art  of  Conjecture,  Applied  to  Questions  of 
Law,"  which  was  printed  in  1709.  Daniel  Bernoulli,  his  brother, 
followed  this  in  1738  with  his  "Specimens  of  a  New  Theory  Con- 
cerning the  Measurement  of  Chance." 

In  Great  Britain,  De  Moivre  reigned  supreme  in  this  branch 
of  applied  mathematics  for  many  years.  He  enjoyed  great  repute 
and,  as  we  shall  see  when  we  discuss  actuarial  science  itself,  he  was 
in  great  demand  to  put  into  practice  his  own  theories,  in  the  form 
of  computations  of  probabilities  of  various  sorts.  But  in  1740  a 
rival  arose,  one  Thomas  Simpson,  a  self-taught  mathematician, 
who  had  also  been  a  necromancer — a  trade  by  this  time  out  of 
fashion  in  connection  with  mathematics — and  who  at  first  enjoyed 
no  standing  whatever,  although  before  his  life  closed  his  accom- 
plishments were  such  that  the  Encyclopedia  Britannica  gives  him 
credit  as  the  greatest  non-academic  mathematician  that  Great 
Britain  has  produced.  Simpson's  first  book,  published  in  1740,  was 
entitled  "The  Nature  and  the  Laws  of  Chance."  It  was  written 
in  language  as  simple  as  the  author  could  employ,  because  it  was 
his  purpose  to  render  the  subject  popular  and  easy  of  comprehen- 
sion by  the  ordinary  untrained  mind.  This  of  itself  was  a  twofold 
offense  against  De  Moivre,  because  it  was,  in  the  view  of  the  dis- 
tinguished mathematician,  both  an  attempt  to  dispute  his  unri- 
valed position  and  also  an  attempt  to  cheapen  and  lower  his  sci- 
ence.   Consequently  the  younger  man,  who,  as  events  have  proved, 


10  DEVELOPMENT  OF 

deserves  more  credit  for  the  growth  and  extension  and  perfection 
of  actuarial  science  and  practice  than  even  De  Moivre,  was  by  no 
means  welcomed  by  his  great  contemporary. 

In  France  the  discussion  of  probabilities  yet  turned  upon  par- 
ticular hazards  in  games  of  chance.  Thus,  so  late  as  1751,  we  find 
the  great  Euler,  one  of  the  most  wonderful  mathematicians  of  all 
time,  sending  forth  a  pamphlet  entitled  "Calculations  of  the 
Probabilities  of  the  Game  of  Eencontre."  The  time  when  France 
would  again  make  a  great  contribution  to  the  science  of  probabili- 
ties was  thus  yet  a  while  deferred. 

In  1773  La  Grange  began  a  work  on  "An  Article  on  Probabil- 
ity," but  it  was  many  years  before  the  subject  was  treated  exhaust- 
ively by  a  Frenchman,  which  seems  all  the  more  remarkable  since 
France  had  so  many  able  mathematicians,  and  also  since  the  first 
work  of  importance  upon  the  subject,  Pascal's  letters,  appeared 
in  that  tongue.  But  the  slowness  with  which  the  science  of  proba- 
bilities in  general  developed  there  may,  as  we  have  seen,  be  as- 
cribed with  much  plausibility  to  the  circumstance  that  Pascal's 
letters  were  concerning  certain  peculiar  gambling  hazards.  This 
view  is  borne  out  by  the  fact  that,  while  discussions  of  probabilities 
in  general  were  missing,  discussions  upon  particular  problems  of 
gambling  were  frequent. 

Between  1750  and  1760  three  Englishmen  contributed  short 
but  valuable  works  upon  the  subject,  Hoyle  in  1754,  James  Dodson 
in  1755,  and  Samuel  Clark  in  1758.  In  1769  John  Bernoulli, 
grandson  of  John  Bernoulli,  the  elder,  and  grand  nephew  of  Jacob 
Bernoulli,  published  a  pamphlet,  the  last  to  bear  the  distinguished 
name  which  was  attached  to  the  first  comprehensive  work  upon 
the  subject. 

France,  which  had  so  long  dragged  behind  Great  Britain  in 
developing  this  science,  began  to  bestir  herself  in  the  last  years 
of  the  eighteenth  century  when,  by  reason  of  the  impulse  which 
the  revolution  gave  all  men,  sciences  and  arts  began  to  develop 
as  never  before.  The  first  fruit  of  this  travail  was  a  pamphlet 
by  Condorcet  in  1785.  But  it  reached  its  full  development  when, 
in  1812,  La  Place's  "Analytical  Theory  of  Probabilities"  appeared. 
This  book  remained  the  classic  text-book  upon  the  subject  for 
many  years,  and  really  exhausted  all  that  was  known  concerning  it. 


INSURANCE  MATHEMATICS  11 

besides  containing  much  that  was  original  and  of  great  value.  It 
remains  to  this  day  one  of  the  three  or  four  really  valuable  books 
upon  the  science. 

Well  might  it  be  such,  also,  for  not  only  was  La  Place  one  of 
the  greatest  mathematicians  of  his  own  or  of  any  time,  but  he 
labored  under  an  encouragement  which  had  never  before  been 
given  any  student  of  the  subject.  Pascal's  letters  brought  forth 
ridicule  and  abuse,  almost  without  a  dissenting  voice,  excepting 
from  advanced  mathematicians  who  were  as  much  out  of  favor  as 
himself.  Much  the  bold  speculator  cared  about  that,  however; 
but  his  successors  did  care.  For,  be  it  known,  the  heterodoxy  of 
to-day  becomes  the  orthodoxy  of  to-morrow  as  the  world  ad- 
vances. When  Pascal's  work  took  its  place  in  the  accepted  views 
of  men  of  learning  the  orthodox  set  their  line  just  beyond  what 
Pascal  had  demonstrated.  All  beyond  that  was  heterodox,  and 
therefore,  as  ever,  the  ban  was  put  upon  the  innovator.  In  France, 
too,  there  was  rather  more  of  this  than  elsewhere,  though  Jacob 
Bernoulli's  great  work  could  not  see  the  light  until  its  author  was 
dead.  In  England  alone,  thanks  to  the  mathematical  turn  which 
Newton's  great  achievements  gave  academical  education,  the  re- 
ception of  the  new  learning  was  warmer,  but  even  there  the  one- 
time innovator,  De  Moivre,  threw  all  sorts  of  obstacles,  not  spar- 
ing ridicule,  in  the  way  of  Simpson. 

With  La  Place  it  was  different.  Not  only  had  the  revolution 
broken  up  the  theological  and  scholastic  prejudices  in  France  and 
installed  instead  a  passion  for  innovation,  but  La  Place  received 
the  direct  support  of  the  emperor,  was  rewarded  directly  for  his 
services,  and  saw  his  book  published  with  honors  and  at  the  na- 
tion's expense.  In  turn  he  dedicated  it  to  the  emperor  who  thus 
befriended  him  and  encouraged  science.  Besides,  La  Place  also 
had  access,  of  course,  to  all  that  had  already  been  written  upon 
the  subject,  which  had  now  been  under  the  consideration  of  some 
of  the  brightest  minds  for  more  than  a  century  and  a  half,  and 
had  been  an  established  and  demonstrated  science  for  almost 
exactly  a  century;  for  Bernoulli's  "Ars  Conjectandi"  appeared  in 
1713,  and  La  Place's  "Analjiical  Theory  of  Probabilities"  in  1812. 

Although  after  the  publication  of  La  PlacS's  monumental  work 
the  evolution  of  the  science  of  probabilities  may  be  considered  to 


12  DEVELOPMENT  OF 

have  been  completed  in  all  essential  particulars,  there  remained 
much  to  do  in  the  matter  of  rendering  the  same  lucid  and  readily 
intelligible,  and  also  of  adapting  it  to  the  practical  purposes  for 
which  it  was  intended.  This  was  the  task  of  the  nineteenth  cent- 
ury especially,  as  will  more  fully  appear  when  we  take  up  actuarial 
science  as  a  separate  study,  although  this  development  also  began 
early  in  the  eighteenth  century.  Most  of  these  productions  have 
taken  the  form  of  works  upon  special  applications  of  the  science  of 
probabilities,  but  there  have  been  at  least  three  books  upon  proba- 
bilities which  deserve  particular  mention. 

In  1838  Professor  Augustus  De  Morgan  published  his  "Essay 
on  Probabilities,"  which,  notwithstanding  the  modesty  of  the  title, 
was  a  very  thorough  and  complete  work  upon  the  subject,  making 
a  thick  volume.  It  became  the  standard  text-book  upon  probabili- 
ties in  the  English  language,  and  held  that  place  until  this  branch 
of  mathematics  began  to  be  taught  in  the  universities  as  one  of  the 
higher  branches,  when  a  work  that  is  more  technical,  less  diffuse, 
with  more  formulas  and  less  of  explanation  and  reasoning,  and, 
above  all,  with  many  examples  for  solution,  was  demanded,  and 
was  supplied  in  Whitworth's  "Choice  and  Chance,"  published  with- 
in the  present  generation,  and  now  the  leading  English  text-book 
upon  the  subject.  In  America,  although  in  our  country  the  princi- 
ples of  the  science  of  probabilities  have  been  more  widely  applied 
in  practical  affairs  than  in  any  other  country,  nothing  has  been 
done  toward  making  text-books  upon  the  subject,  and  very  little 
instruction  upon  it  has  been  given  in  schools  or  colleges. 

A  work  of  the  greatest  importance  upon  the  principles  of  the 
science  of  probabilities,  and  upon  their  application  to  many  things, 
was  published  in  French  in  1845.  It  was  the  work  of  a  Belgian, 
M.  Quetelet,  and  was  entitled  "Letters  on  the  Theory  of  Proba- 
bilities as  Applied  to  Moral  and  Political  Sciences."  The  writing 
of  these  letters  began  in  1837,  without  any  thought  of  publishing 
the  same.  Their  value  and  significance  may  be  estimated,  however, 
by  the  fact  that  almost  immediately  upon  their  appearance  thej 
were  translated  into  English  and  other  languages,  and  became  the 
most  valuable  text-book  upon  the  subject  that  its  students  could 
turn  to.  This  was  very  largely  because  of  the  fullness  and  clear- 
ness with  which  the  application  of  the  law  of  probabilities  to  point 


INSURANCE  MATHEMATICS  13 

the  lessons  of  statistics  was  set  forth,  though  the  same  lucidity- 
marked  the  whole  of  the  work. 

The  history  of  the  development  of  the  science  has  thus  far 
been  traced  from  the  earliest  times  to  the  present.  It  is,  as  we  have 
seen,  essentially  a  purely  modern  science.  This  branch  of  mathe- 
matics could  hardly  develop  in  advance  of  algebra,  decimal  nota- 
tion, decimal  fractions,  series,  annuities  certain  and  the  like,  for 
much  of  it,  in  theory  or  application,  depends  upon  one  or  all  of 
these.  Singular  is  it,  also,  that  the  first  great  work  upon  the  sub- 
ject and  the  last,  in  the  sense  of  an  original  and  suggestive  work, 
the  contributions  of  Pascal  and  Quetelet,  have  both  been  in  the 
form  of  letters,  written  with  no  intention  originally  of  publishing 
the  same.  We  shall  see,  when  we  come  to  consider  the  origin  of 
mortality  tables  and  the  evolution  of  actuarial  science,  that  the 
growth  of  the  body  of  knowledge  concerning  probabilities  has  been 
accompanied  by  the  application  of  the  principles  of  the  science  to 
the  practice  of  insurance,  and  especially  of  life  insurance. 

It  remains,  then,  only  to  discuss  what  this  science  of  probabili- 
ties is  which  we  have  been  tracing  through  its  different  stages  of 
historic  development.  The  primary  discovery  was  that  when  a 
thing  had  happened  to  a  certain  number  out  of  a  group,  to  each 
of  which  it  was  a  priori  equally  likely  to  happen,  the  probability 
that  it  had  happened  to  a  particular  one  in  the  group,  when  one 
did  not  know  which,  could  be  expressed  by  a  fraction,  of  which 
the  number  to  whom  it  had  happened  was  the  numerator,  and  the 
number  in  the  whole  group,  that  is,  the  number  to  whom  it  had 
happened  and  to  whom  it  failed  to  happen,  combined,  was  the 
denominator.    This  was  the  fundamental  discovery. 

Following  this  at  no  great  distance  was  the  discovery  that  the 
chance  that  the  thmg  had  failed  to  happen  to  a  particular  one  in 
the  group  could  be  measured  by  a  fraction,  with  the  number  to 
whom  it  had  failed  to  happen  as  a  numerator,  and  with  the  same 
denominator  as  before.  A  little  consideration  sufl&ced  to  show  that 
these  were  complementary  values,  the  sum  of  which  was  unity, 
because  adding  them  gave  the  number  to  whom  it  happened  plus 
the  number  to  whom  it  did  not  happen,  both  for  numerator  and  for 
denominator.  Thus  it  was  determined  that  the  sum  of  the  chances 
that  a  thing  happens  and  that  it  did  not  happen  is  always  unity; 


14  DEVELOPMENT  OF 

but,  since  such  a  summation  also  brings  certainty,  for  as  to  a  given 
individual  it  may  with  confidence  be  affirmed  that  the  thing  either 
has  happened  or  has  not  happened,  it  has  been  customary  to  rep- 
resent the  mathematical  value  of  certainty  by  unity.  A  very  in- 
genious proof  that  this  is  the  correct  thing,  and  indeed  the  only 
thing  to  do,  is  given  by  Whitworth  in  "Choice  and  Chance,"  already 
referred  to;  it  will  be  seen  that  actual  proof  offers  difficulties, 
though  it  is  not  hard  to  see  that  the  thing  is  true.  The  following 
is  the  passage : 

"But  it  may  be  asked,  is  certainty  a  degree  of  probability  at  all, 
or  can  smaller  degrees  of  probability  be  said  to  have  any  ratio  of 
certainty?  Yes.  For  if  we  refer  to  the  instance  already  cited  of 
the  six  passengers  in  the  ship  (from  which  one  person  had  been 
lost)  we  observe  that  the  chance  of  the  lost  man  being  a  passenger 
is  six  times  as  great  as  the  chance  of  his  being  our  friend  (one 
of  the  six).  This  is  the  case,  however  great  our  ignorance  of  the 
circumstances  of  the  event ;  and  it  will  evidently  remain  true  until 
we  attain  some  knowledge  which  affects  our  friend  differently  from 
his  fellow  passengers.  But  the  news  that  the  lost  man  was  a  pas- 
senger does  not  affect  one  passenger  more  than  another.  There- 
fore, after  receiving  this  news,  it  will  still  hold  good  that  the 
chance  of  the  lost  man  being  a  passenger  is  six  times  as  great  as 
the  chance  of  its  being  our  friend.  But  it  is  now  certain  that  the 
lost  man  was  a  passenger;  therefore  the  probability  that  it  was  our 
friend  is  one-sixth  of  certainty.'^ 

Since  by  the  fundamental  rule  of  probabilities  we  will  also  find 
that  it  is  one-sixth  of  unity,  it  follows  that  certainty  is  correctly 
represented  by  unity. 

From  these  simple  laws  the  science  of  probability  extended  to 
compound  probabilities.  One  of  the  most  remarkable  of  the  early 
discoveries  was  that,  when  you  know  the  chance  that  one  thing 
will  happen,  and  also  the  chance  that  another  will  happen,  the 
chance  that  both  will  happen,  unless  one  is  prohibitive  of  the 
other,  is  not  the  sum  of  the  chances,  but  their  product.  And  from 
this  the  investigations  proceeded  to  deal  with  all  imaginable  com- 
pound and  complex  probabilities,  many  of  the  most  difficult  being 
treated  in  advance  of  the  development  of  principles  for  determin- 
ing the  simpler  and  easier  values. 


INSURANCE  MATHEMATICS  15 

It  was  a  long  step,  but  one  soon  taken,  from  the  probability 
that  a  thing,  known  to  have  happened  to  one  of  a  group,  has 
happened  to  a  particular  one,  to  the  probability  that  events  will 
occur  in  future,  a  probability  based  upon  averages,  depending 
upon  their  reliability,  their  extent,  the  correctness  of  classifi- 
cation and  their  applicability.  These  problems  have  been  so  diffi- 
cult in  operation  that  to  this  day  an  enormous  amount  of  the 
theory  of  probabilities  remains  unappropriated  by  applied  sciences 
because  of  the  great  practical  difficulties  in  the  way  of  so  apply- 
ing them.  Many  very  interesting  subjects  of  speculation  in  the 
laws  of  probabilities  are  therefore  neglected  by  the  actuary  in 
his  practice,  and,  indeed,  only  the  simpler  elements  of  the  science 
are  at  the  present  time  employed  by  him,  though  he  would  be  a 
rash  man  indeed  who  would  undertake  to  say  what  will  be  the 
limit  of  their  practical  application  next  year. 

Though  coming  so  late  in  the  family  of  mathematical  sciences, 
probabilities  already  rejoice  in  a  biographer  of  no  mean  reputation. 
Todhunter,  the  great  English  mathematician,  has  written  and 
published  an  extensive  work,  now  unhappily  out  of  print,  upon 
the  history  of  probabilities,  in  which  he  traces  the  growth  of  the 
true  theories,  the  correction  of  errors  and  the  development  of 
new  principles  with  great  fullness  and  precision.  His  account 
begins  with  the  earliest  writers  upon  the  subject  and  closes  with 
La  Place.  From  it  much  of  the  material  for  this  lecture  has  been 
obtained,  and  particularly  so  concerning  some  of  the  smaller  and 
less  important  pamphlets  which  are  not  to  be  found  in  the  usual 
library  of  the  actuary.  As  to  the  more  important,  including  even 
Bernoulli's  "Ars  Conjectandi"  and  De  Moivre's  'TDoctrine  of 
Chances,"  as  well  as  La  Place's  great  work,  references  to  such 
have  been  made  directly. 


16  DEVELOPMENT  OF 


SECOND  LECTURE— MORTALITY  TABLES,  ANNUITIES 
CERTAIN  AND  ACTUARIAL  SCIENCE  PROPER. 

We  have  seen  how  the  fundamental  mathematics,  requisite 
before  actuarial  science  could  develop,  evolved;  we  have  now  to 
consider  the  genesis  and  evolution  of  the  application  of  the  laws 
of  probability  to  insurance,  and  especially  to  life  insurance.  And 
we  shall  find  that,  first  of  all,  there  needed  to  be  something  for 
the  law  of  probability  to  act  upon,  viz.,  a  mortality  table,  and  also 
a  handmaiden,  in  the  form  of  a  developed  law  of  compound  in- 
terest and  discount  and  of  annuities  certain. 

First,  then,  as  to  mortality  tables.  We  find  the  first  of  them 
in  times  just  back  of  mediaeval,  and  just  this  side  of  ancient.  The 
pretorian  prefect,  Ulpian,  in  364  A.  D.,  put  forth  a  table  of  life 
expectancies  or  probable  lives,  invented  for  the  purpose  of  esti- 
mating the  value  or  annuities  for  probate  purposes.  The  Eoman 
law  at  that  time  provided  that  not  more  than  a  certain  portion 
of  the  value  of  an  estate  could  be  willed  away  from  the  heirs. 
But  testators  left  annuities,  as  charges  upon  their  estates,  the 
same  being  'payable  to  beneficiaries  for  life ;  and  so  it  became 
necessary  to  estimate  the  value  of  these  annuities.  The  Ulpian 
method  was  simple.  It  assumed  that  a  term  of  an  annuity  cer- 
tain could  be  employed  to  express  the  value  of  a  life  annuity,  and 
that  this  term  was  that  of  the  probable  life  of  the  annuitant,  or 
that  term  of  years  at  the  end  of  which  the  chances  would  be 
equal  that  he  was  living  or  dead. 

Actuarial  science  could  have  no  real  genesis  until  this  fallacy 
was  dead  and  the  truth  had  appeared.  Ulpian's  experiment  was 
forgotten,  and  his  theory  was  wholly  lost  when  we  next  encounter 
something  like  a  mortality  table,  and  this  time  with  true  princi- 
ples applied  to  ascertain  the  value  of  an  annuity.  The  author  was 
no  less  a  person  that  John  de  Witt,  Grand  Pensionary  of  Holland 
and  West  Friesland,  who  turned  aside  from  the  onerous  duties 
of  state  to  consider  the  subject  of  life  pensions  or  annuities.  Such 
were  already  granted  by  states  which  were  hard  pressed  for  money 
and  which  could  not  negotiate  loans  in  the  usual  way.  De  Witt 
formed  the  idea  that  the  prices  at  which  such  were  sold  were  ut- 


INSURANCE  MATHEMATICS  17 

terly  inadequate,  and  he  proceeded  to  investigate  the  subject. 
His  work  was  done  before  1671,  when  the  results  of  his  labors 
were  presented  in  a  state  paper.  This  was  not  many  years  after 
the  appearance  of  the  first  pamphlets  about  probabilities,  as,  for 
instance,  Pascal's  letters.  It  was  more  than  forty  years  before 
the  publication,  in  1712,  of  Jacob  Bernoulli's  "Ars  Conjectandi," 
the  first  comprehensive  work  on  probabilities.  Therefore,  it  may 
be  truly  said,  not  merely  that  actuarial  science  began  with  de 
Witt,  but  also  that  he  built  much  of  the  substructure  for  it ;  since 
in  his  work  he  developed  newly,  if  not  first  of  all,  the  fundamental 
principles  of  probabilities. 

Most  important  of  all  else,  though,  in  his  work,  was  the  dis- 
covery of  the  true  theory  of  the  operation  necessary  to  compute 
the  values  of  annuities;  and  this  was:  Take  a  large  number  of 
annuitants  at  a  given  age,  the  mortality  for  each  age  being  known, 
and  follow  them  down  through  life  as  they  drop  out  and  disap- 
pear, computing  the  amount  which  would  need  to  be  paid  each 
year;  reduce  these  sums  to  their  present  value;  add  them  all 
together,  and  you  have  the  aggregate  present  value  of  all  sums 
payable  on  account  of  these  annuities.  The  equal  share  of  each 
annuitant  in  this  aggregate  is  manifestly  the  value  of  his  annuity. 

The  value  thus  found  is  not  the  same  that  would  be  found  by 
treating  the  annuity  as  equivalent  to  an  annuity  certain  for  the 
term  of  the  expectancy,  or  of  probable  life,  according  to  the  same 
table  of  mortality.  But  this  value  will  prove  by  actually  running 
out,  on  the  basis  of  the  interest  and  mortality  assumptions,  while 
the  value  from  the  annuity  certain  for  a  term  equivalent  to  the 
expectation  or  the  probable  life  will  not.  Moreover,  when  we 
come  to  examine  into  the  matter,  we  cannot  fail  to  be  impressed 
with  the  circumstance  that  the  expectation  could  only  furnish  an 
equivalent  when  there  is  no  interest  counted  upon,  for  it  is  the 
average  of  years  of  life ;  while  in  effect,  the  value  of  the  annuity  is 
the  average  of  the  same  years  of  life,  discounted.  But  the  dis- 
count is  greater  for  payments  which  are  longer  deferred;  from 
which  fact  it  appears  that  the  equivalence  no  longer  holds,  when 
discount  is  introduced. 

The  Ulpian  theory  died  hard.  It  reappeared  repeatedly  after 
actuaries  had  learned  to  deal  with  the  subject  according  to  science 


18  DEVELOPMENT  OF 

and  reason.  Thus  in  1738,  after  the  true  principles  were  clearly- 
enunciated  and  had  been  adopted  by  mathematicians  generally, 
a  learned  London  barrister,  Weyman  Lee,  published  an  ingenious 
book  in  which  he  undertook  to  prove  the  equivalence.  The  name 
of  his  book  was  "An  Essay  to  Ascertain  the  Value  of  Leases  and 
Annuities  for  Years  and  Lives."  The  following  passage  is  illus- 
trative of  his  reasoning: 

"I  suppose  no  one  will  controvert  these  points :  That  he  who 
has  an  annuit}^  for  the  life  of  a  person,  has  an  annuity  for  such  a 
term  of  years  as  such  person  in  fact  shall  live;  and,  when  he  buys 
it,  the  term  of  years  to  which  any  person's  life  shall  be  prolonged 
being  uncertain,  that  he  buys  it  for  such  a  time  as  there  is  a 
chance  or  reasonable  probability  that  the  person  may  live  whose 
life  is  nominated." 

Lee  also  objected  that  the  other  theory  made  the  value  of 
an  annuity  by  the  same  mortality  table  equivalent  to  the  values 
for  different  terms  of  years,  according  as  the  interest  assumed 
was  one  rate  or  another.  We  have  seen  that  this  is  necessarily 
so,  and  that  only  when  no  interest  is  assumed  is  the  value  equal 
that  of  an  annuity  certain  for  the  expectancy. 

The  false  theory,  dating  back  to  Ulpian,  and  so  ably  cham- 
pioned by  Lee,  is  a  very  natural  generalization  when  one  makes 
a  hasty  conclusion,  and  it  is  certainly  extremely  plausible.  As  if 
to  emphasize  that  the  human  mind  in  its  evolution  ever  travels 
over  the  same  track,  we  find  this  idea  reappearing  every  now  and 
then  by  a  strange  sort  of  atavism,  as  it  were,  and  in  the  most  un- 
expected quarters.  Thus  within  the  year  a  leading  financial  paper 
of  Xew  York  gravely  announced  editorially: 

"Life  insurance  companies  employ  two  factors  in  fixing  pre- 
miums on  policies  of  insurance.  They  are,  first,  the  average  ex- 
pectation of  life  of  the  insured  person,  according  to  carefully  pre- 
pared mortality  tables;  second,  the  rate  of  interest  at  which 
money  can  be  invested.  The  company  takes  the  case  of  a  man  at, 
say,  21  years  of  age,  and  finds  from  the  mortality  tables  that  his 
average  expectation  of  life  is,  say,  30  years.  By  simple  arith- 
metical calculations  it  computes  the  amount  of  money  that  such 
a  man  should  pay,  annually,  semi-annually,  or  quarterly,  as  the 
case  may  be,  in  order  that  his  money  at  the  assumed  rate  of  in- 


INSURANCE  MATHEMATICS  19 

terest,  say  3J  per  cent,  shall  at  least  equal  the  amount  of  his  policy 
at  the  end  of  thirty  years.  The  amount,  thus  computed,  with  an 
allowance  for  expenses  and  profit,  is  his  premium." 

This  apparent  digression  is  really  no  digression  at  all.  We 
are  to  follow  the  growth  of  actuarial  science,  and  it  is  worth 
while,  therefore,  to  know  what  it  is,  and  particularly  what  it  is 
not,  and  why,  therefore,  John  de  Witt  must  be  considered  its 
founder,  and  not  Ulpian. 

de  Witt,  in  addition  to  discovering  the  fundamentally  true 
theory  of  annuities  for  life,  and  applying  the  principles  of  prob- 
abilities and  compound  discount  to  the  ascertainment  of  their 
values,  also  invented  a  mortality  table.  This  he  constructed  on 
a  plan  of  equal  decrements,  limited  to  from  ages  three  to  fifty, 
however,  followed  by  other  periods  in  which  the  law  of  equal 
decrement  was  also  supposed  to  work.  We  shall  encounter  this 
supposed  law,  and  modifications  of  it,  again  and  again.  It  reap- 
peared in  the  work  of  the  earliest  English  actuary  who,  so  far 
as  we  know,  was  not  acquainted  with  de  Witt's  work  at  all.  De 
Witt  also  foresaw  that  there  would  be  a  selection  of  the  best  lives 
for  annuitants,  and  that  a  mortality  much  lower  than  in  the  gen- 
eral population  was  to  be  expected.  Thus  he  foresaw  a  truth 
which  it  took  the  British  government  many  years,  involving  great 
loss,  to  discover. 

The  next  discovery  was  by  the  British  philosopher,  Halle}^, 
who,  in  1693,  published  in  the  "Transactions"  of  the  Eoyal  So- 
ciety the  result  of  his  investigations  into  the  mortality  of  the  city 
of  Breslau,  Germany,  which  had  given  out  a  complete  statement 
of  deaths,  by  ages,  for  a  number  of  years.  Assuming  a  stationary 
population,  Halley  proceeded  to  construct  a  mortality  table  from 
the  records  of  deaths  only.  Halley  thus  produced  a  complete 
mortality  table  showing  the  diminution  of  the  number  living  each 
year  out  of  an  original  group.  By  means  of  this  table  he  also 
proceeded  to  give  the  now  common  and  always  simple  formulas 
for  computing  the  values  of  annuities.  The  table  which  he  made 
is  the  first  that  was  made  from  population  statistics.  The  develop- 
ment of  actuarial  science  was  in  a  large  degree  due  to  him,  de 
Witt's  work  lay  buried  in  the  archives  of  Holland  and  West  Fries- 
land,  and  was  only  uncovered  within  the  last  fifty  years. 


20  DEVELOPMENT  OF 

I  have  said  that  Halley's  Breslau  table  was  the  earliest  that 
was  drawn  from  actual  experience,  and  this  is  true  so  far  as  a 
table  showing  the  mortality  at  each  year  of  age  is  concerned.  But, 
as  early  as  1662,  John  Graunt  had  published  an  analysis  of  the 
mortuary  returns  of  London,  and  had  formulated  a  mortality  table 
in  a  rough  form,  but  not  giving  the  rate  of  mortality  at  each  year 
of  age.  His  work,  however,  had  attracted  much  attention,  and 
passed  through  several  editions. 

The  next  great  advance  was  when  De  Moivre  published  in  1725 
his  great  work  on  life  annuities.  A  French  exile,  living  in  London,, 
he  had  already  distinguished  himself  in  mathematics,  and  es- 
pecially in  his  works  on  probabilities.  Now  he  proceeded  to  apply 
his  learning  to  the  practical  purpose  of  determining  the  values- 
of  leases  for  life  and  other  life  annuities.  Already  the  summa- 
tion of  series,  when  following  some  mathematical  law,  had  been 
developed  to  a  point  which,  he  plainly  saw,  rendered  it  compara- 
tively simple  and  easy  to  compute  annuity  values,  whether  for  one 
or  for  more  lives,  if  only  a  mortality  table  could  be  brought  to 
follow  a  mathematical  law.  He  investigated  the  Breslau  table 
put  forth  by  Halley,  and  discovered  that  within  reasonable  limits, 
for  a  long  period  of  life  at  least,  and  possibly  for  the  whole,  the 
number  surviving  to  each  year  of  age  out  of  a  given  number  set- 
ting out  from  the  earliest  age  of  the  table,  was  a  term  in  a  de- 
creasing arithmetical  series.  That  is,  he  found  that  if  the  number 
surviving  be  treated  as  a  term  of  such  a  series,  it  would  diminish 
very  nearly  as  in  the  original  table.  This  means  that  the  num- 
ber of  deaths  could  be  assumed  to  be  the  same  for  each  year  of  age 
out  of  the  number  originally  setting  out.  Thus,  if  the  deaths, 
for  instance,  were  1,000  per  annum  out  of  an  original  100,000,. 
setting  out  at  age  20,  this  would  mean  1,000  out  of  100,000  at 
age  20,  1,000  out  of  99,000  at  age  21,  etc.  So  the  probabili- 
ties of  death  would  each  year  increase,  not  because  of  any  change 
in  the  numerator  of  the  fraction,  but  solely  because  of  the  steady 
and  regular  diminution  of  the  denominator.  Without  going  into- 
details  at  this  time,  it  must  be  readily  seen  that  for  the  purpose 
of  determining  any  value  or  function  whatever,  the  arrangement 
of  a  mortality  table  as  a  mere  regularly  decreasing  arithmetical 
series  offered  great  advantages. 


INSURANCE  MATHEMATICS  21 

De  Moivre  also  discovered,  somewhat  later  in  his  career,  that 
complete  tables  of  annuities  could  be  made  by  a  continuous  proc- 
ess, starting  from  the  extreme  limit  of  life,  without  computing 
all  the  values  separately  which  go  to  make  up  the  value  of  each 
annuity.  This  discovery  applied,  first,  to  annuities  certain;  but 
its  application  to  life  annuities  was  soon  discerned  as  well.  A 
word  of  explanation  may  be  needed  to  make  the  matter  clear  to 
you,  and  I  linger  to  give  it  because  this  is  the  first  great  labor- 
saving  invention  in  the  science,  which  now  embraces  so  many. 

Suppose,  then,  that  you  know  the  value  of  one,  due  at  the  end 
of  one  year,  and  desire  to  discover  the  value  of  an  annuity  of  one 
for  two  years.  This  is  plainly  worth  at  the  end  of  one  year,  and 
before  the  first  annuity  payment  is  made,  one  plus  the  value  of 
one,  due  at  the  end  of  one  year.  Therefore,  at  the  beginning 
of  the  first  year  it  will  be  worth  one  plus  the  present  value  of 
one,  due  at  the  end  of  one  year,  the  whole  multiplied  by  the  pres- 
ent value  of  one,  due  at  the  end  of  one  year.  And  so,  by  a  con- 
tinuation of  this  process,  a  complete  table  of  the  values  of  an- 
nuities for  each  of  long  periods  may  be  successively  computed. 
The  same  principle,  the  discounts  merely  being  made  also  to  in- 
clude the  probability  as  well  as  the  interest  factor,  applies  to  life 
annuities;  and  this  was  De  Moivre's  discovery.  It  was  also  inde- 
pendently seen  by  both  the  great  mathematician  Euler  and  also 
by  Thomas  Simpson,  De  Moivre's  younger  rival  in  England. 

It  was  in  1742  that  the  first  work  of  Thomas  Simpson  upon 
the  subject,  his  "Doctrine  of  Annuities  and  Reversions,"  made  its 
appearance.  Simpson,  as  was  stated  in  the  previous  lecture,  was 
a  self-taught,  non-academic  mathematician,  believed  to  be  the 
greatest  of  such  that  England  has  produced.  He  had  but  emerged 
from  obscurity  and  ill  fame,  owing  to  conjuring  and  necromancy. 
That  he  should  come  forth  as  a  rival  of  De  Moivre  was  almost 
insupportable  by  the  latter,  who  was  now  in  the  flower  of  his 
age  and  at  the  zenith  of  a  well-earned  reputation.  Simpson's 
book,  too,  was  an  attempt  to  popularize  the  science,  never  a  popu- 
lar movement  among  those  who  hope  to  profit  by  keeping  it  ex- 
clusive. And,  worst  of  all,  he  had  little  respect  for  the  doctrine 
of  equal  decrements,  and,  indeed,  did  not  spare  it. 

Simpson  urged  a  return  to  Halley's  method  of  taking  the  facts 


22  DEVELOPMENT  OF  . 

as  one  found  them,  instead  of  seeking  to  compel  the  mortality- 
table  to  follow  a  rule  to  which  it  was  indifferently  suited.  At  the 
same  time,  on  the  basis  of  De  Moivre's  principle,  he  discovered 
an  equal  age  formula  for  joint  life  computations,  which  greatly- 
simplified  the  operations,  and  which  bears  his  name  to  this  day. 
But  perhaps  his  greatest  distinction  is  this,  that  he,  first  of  all 
actuaries,  fully  realized  the  importance  of  the  life  insurance  phase. 
Up  to  his  day  life  insurance  was  known  in  but  two  forms,  one 
being  policies  for  short  terms,  without  privilege  of  renewal,  and 
the  other,  insurance  for  the  whole  of  life  for  indeterminate 
amounts,  to  be  determined  by  the  proceeds  of  assessments.  Simp- 
son introduced  the  idea  of  whole  life  insurance  for  level  annual 
premiums  and,  as  we  shall  see,  he  also  afterward  introduced  the 
practice. 

At  about  the  same  time  Des  Parcieux,  the  elder,  produced  in 
France  his  work,  "Probabilities  of  Human  Life,'^  published  in 
1746.  This  was  practically  the  beginning  of  annuities  in  that 
country.  It  was  followed  in  1749  by  Buffon's  "Tables  of  Prob- 
abilities of  Human  Life,"  which  was  mainly  a  discussion  of  tables 
already  published,  but  included  also  some  new  tables.  Des  Par- 
cieux, in  his  work,  gave  tables  which  he  had  constructed  from 
the  experience  of  two  tontine  funds  and  from  the  mortality  in 
fourteen  monasteries.  The  last  became  the  standard  table  for 
insurance  in  France  and  remained  standard  until  within  the  last 
generation. 

In  1753  the  scientist  Kersseboom  of  The  Hague  brought  out 
a  volume  discussing  the  tables  of  Halley  and  Buffon,  and  develop- 
ing some  of  the  principles  of  the  science. 

James  Dodson,  an  English  university  man  of  high  standing, 
began  in  1748  the  preparation  of  a  volume  entitled  "Analytical 
Solution  of  Problems  Eelating  to  Annuities."  Mr.  Dodson  began 
to  take  an  interest  in  the  subject  comparatively  late  in  his  busy 
life,  and  his  interest  was  rather  sharpened  than  otherwise  by  the 
discovery  that,  while  he  might  procure  life  insurance  at  very  high 
rates  in  a  stock  company,  and  for  a  very  short  period,  he  could 
get  no  whole  life  insurance  at  all,  even  in  the  Amicable  on  the 
assessment  plan,  because  he  was  more  than  45  years  of  age.  He 
became  one  of  the  earliest  and  most  active  and  influential  of 


INSURANCE  MATHEMATICS  23 

Simpson's  converts  to  the  view  that  life  insurance  could  be  fur- 
nished on  a  whole  life  plan  with  equal  annual  premiums,  and 
that  safe  premiums  for  the  same  could  he  computed.  Together 
these  men  labored  for  several  years  to  bring  enough  people  to- 
gether to  organize  a  company  on  the  mutual  plan — they  had  no 
hope  of  interesting  stockholders  in  so  long  and  uncertain  a  venture, 
even  in  those  speculative  days.  The  result  was  the  "Society  for 
the  Equitable  Assurance  of  Lives  and  Survivorships,"  which, 
now  generally  known  as  "The  Old  Equitable,"  survives  to  this  day. 
The  organization  was  not  without  difficulties,  and  had  to  be  ef- 
fected as  an  unincorporated  voluntary  association,  practically  a 
copartnership,  because  the  Parliament  refused  a  charter  on  the 
ground  that  the  undertaking  was  very  risky. 

The  tables  which  were  employed  by  the  Equitable  were  based 
upon  a  mortality  table  constructed  by  Dodson  from  the  mortuary 
statistics  of  London.  Fortunately  the  experience  proved  an  ex- 
aggeration of  the  deaths  which  the  company  experienced,  and  as. 
it  did  much  more  life  insurance  than  annuities,  the  result  was 
favorable.  Of  course  it  would  have  been  disastrous  if  the  con- 
trary had  proven  the  case,  or  if  the  annuity  business  had  ex- 
ceeded the  life  insurances,  for  the  same  mortality  table  was  em- 
ployed for  both. 

Neither  Thomas  Simpson  nor  James  Dodson,  though  they  con- 
tributed so  much  to  the  foundation  of  the  Equitable,  and  indeed 
were  and  are  recognized  to  have  been  its  founders,  having  stirred 
up  interest  in  the  matter  by  their  publications  and  by  lectures 
upon  the  subject,  in  London  and  elsewljere,  was  included  in  the 
management  of  the  society;  but  it  set  forth  upon  its  career  on 
plans  devised  and  furnished  by  them,  and  its  success  is  theirs. 

In  the  same  year  that  the  Equitable  began  operations  there 
appeared  a  first  edition  of  a  most  remarkable  contribution  to  act- 
uarial science,  by  Dr.  Eichard  Price,  a  Unitarian  preacher,  who 
was  destined  to  have  much  to  do,  both  indirectly  and  directly, 
with  the  development  of  life  insurance,  and  of  the  Equitable  in 
particular.  The  work  was  entitled  "Observations  on  Reversion- 
ary Payments."  It  passed  into  four  editions,  each  much  en- 
larged, during  the  succeeding  twenty-one  years,  and  was  the  au- 
thoritative work  upon  the  subject  for  a  very  long  time,  a  revised 
edition  being  published  so  late  as  1806. 


24  DEVELOPMENT  OF 

Dr.  Price  warmly  supported  Thomas  Simpson's  view  that  it 
^as  folly  to  construct  mortality  tables  by  purely  mathematical 
principles,  and  urged  that  investigations  of  the  actual  death  rates 
be  made  as  the  only  means  of  making  a  reliable  table  of  mortality. 
He  also  encouraged  with  enthusiasm  the  work  of  the  Equitable, 
and  in  his  later  editions  contrasted  sharply  its  scientific  and  safe 
system,  assuring  solvency,  with  the  unsafe  plans  and  insolvent 
condition  of  all  the  assessment  annuity  concerns  with  which  the 
country  was  then  pestered.  In  addition  to  his  demonstrations  of 
these  things  he  himself  analyzed  several  bodies  of  population 
statistics,  such  as  those  of  Sweden,  which  had  recently  been  pub- 
lished, constructing  mortality  tables  from  them;  and,  which  proved 
most  important  of  all,  he  collected  the  data  concerning  deaths  m 
several  parishes  of  Great  Britain,  notably  Northampton,  and  made 
mortality  tables  from  these.  The  Northampton  table  enjoyed 
great  repute,  and  was  the  standard  for  both  life  insurance  and 
>)nnuity  computations  for  many  years.  It  was  adopted  by  the 
Equitable,  and  Dr.  Price  was  invited  to  prepare  monetary  tables 
on  that  basis  for  that  company.  Later,  his  nephew,  William  Mor- 
gan, was  made  actuary  of  the  company,  a  post  which  he  occupied 
throughout  his  long  life.  The  theories  of  De  Moivre  were,  for 
the  time,  completely  lost  sight  of,  and  it  was  conceded  everywhere 
that  the  only  correct  system  of  developing  a  mortality  table  was 
to  follow  the  facts  of  statistics  closely,  not  expecting  the  same 
to  express  a  mathematical  law.  William  Morgan  also  distinguished 
himself  by  the  publication  of  many  books  and  pamphlets,  includ- 
ing contributions  to  the  Eoyal  Society,  of  which  he  was  an  hon- 
ored member.  His  most  important  work,  perhaps,  was  entitled 
"Doctrine  of  Annuities  and  Assurances  on  Lives  and  Survivor- 
ships," published  in  1779.  To  survivorships  he  devoted  much 
study.  He  also  edited  and  revised  the  edition  of  Eichard  Price's 
"Observations  on  Keversionary  Payments,"  which  appeared  in 
1806. 

In  France  progress  was  also  being  made,  although  there  annui- 
ties proved  more  popular  by  far  than  life  assurances,  which  is 
true  even  to  this  day.  In  1767  the  great  mathematician,  Euler, 
published  a  work  upon  "General  Researches  into  the  Mortality 
and  Increase  of  the  Human  Race,"  and  in  1781  DesParcieux,  the 


INSURANCE  MATHEMATICS  25 

younger,  contributed  a  "Treatise  on  Annuities."  In  Great  Britain, 
Maseres,  with  a  foreign-sounding  name,  published  a  large  book, 
of  no  very  great  value,  entitled  "Principles  of  the  Doctrine  of 
Annuities,"  in  1783;  in  it  he  endeavored  to  make  the  whole  thing 
a  matter  of  arithmetic. 

There  was  great  progress  in  life  insurance  during  the  last  part 
of  the  eighteenth  century,  but  little  was  done  in  the  direction 
of  increasing  either  the  material  upon  which  actuarial  science 
works,  or  the  methods  of  doing  the  work.  The  Northampton  table 
was  now  fully  accepted  in  Great  Britain  as  the  standard,  and 
the  government  even  sold  annuities  based  upon  its  accuracy. 

In  1810  the  monumental  work  of  Francis  Bailly  upon  "Life 
Annuities  and  Assurances"  appeared.  It  was  the  first  compre- 
hensive and  thorough  treatise  that  had  ever  been  published  upon 
the  subject.  What  was  known  had  by  this  time  clarified  in  men^s 
minds,  and  it  was  now  possible  to  treat  the  topic  thoroughly  and 
well.  Bailly's  equipment  for  this  task  was  nearly  as  good  as 
that  of  La  Place  for  dealing  with  probabilities;  and  it  is  note- 
worthy that  the  two  epoch-making  works  made  their  appearance 
at  about  the  same  time.  La  Place's  "Probabilities"  appearing  m 
1812.  Bailly's  book  was  so  good  and  so  comprehensive  that  as 
late  as  1864,  when  by  reason  of  the  invention  of  commutation 
columns  as  a  convenience  for  calculations  and  of  a  change  in  the 
symbols,  the  book  had  become  antiquated,  its  worth  caused  it  to 
be  rewritten,  edited  and  produced  with  modern  symbols,  so  as  to 
continue  to  be  useful  to  the  student.  Yet,  notwithstanding  his 
abilities  and  accomplishments,  some  of  which,  of  course,  em- 
braced the  correction  of  others'  errors,  or  perhaps,  more  accurately 
stated,  because  of  these  things,  Bailly  was  anything  but  welcomed 
by  Morgan,  now  aging,  and  now  also  the  absolute  master  of  things 
actuarial.  In  consequence  a  feud,  now  disguised,  now  breaking 
forth,  raged  between  the  men  during  their  joint  career. 

In  1815  appeared  Joshua  Milne's  ^^aluation  of  Annuities  and 
Assurances,"  also  an  epoch-making  work ;  for,  in  addition  to  much 
that  was  new  and  clearer  in  the  text,  the  author  included  a  new 
mortality  table,  drawn  from  the  experience  of  the  parish  of  Car- 
lisle. This  table  was  the  first  to  be  graduated  so  that  it  seemed 
to  express  a  regular  force  of  mortality,  although  for  the  same  no 


26  DEVELOPMENT  OF 

mathematical  expression  was  known  or  sought.  The  influence  of 
Eichard  Price  was  yet  too  powerful  to  permit  turning  one's  atten- 
tion in  that  direction  anew.  The  graduation  was  by  what  is  now 
known  as  the  graphic  method,  and  it  was  necessary,  particularly  in 
this  case,  because  the  number  of  lives  dealt  with  was  very  small. 
This  table  showed  a  much  lower  mortality  rate  than  the  North- 
ampton table  and  came  far  nearer,  also,  to  expressing  what  the 
companies  were  finding  to  be  their  own  experience.  This  was 
even  confirmed  by  the  experience  of  the  Equitable  itself.  In  con- 
sequence most  of  the  younger  companies  promptly  adopted  the 
new  table,  especially  as  the  Northampton  was  not  merely  not 
fitted  for  annuities  but  was  positively  dangerous.  But  the  North- 
ampton held  its  own  with  the  Equitable  and  some  of  the  older 
companies,  and  there  was  a  bitter  fight  about  the  matter,  the 
echoes  of  which  have,  even  now,  scarcely  ceased  to  rumble. 

In  1820  appeared  Hendry's  work  on  "Life  Annuities,"  and  in 
1825  Griffith  Davies'  remarkable  book,  entitled  "Tables  of  Life 
Contingencies."  This  work  introduced  into  general  use  the  second 
great  labor-saving  device  of  actuarial  science,  the  commutation 
column.  A  suggestion  of  something  of  this  sort  had  been  made 
by  William  Dale  as  early  as  1772  in  his  "Introduction  to  the  Study 
of  Annuities,"  and  it  had  also  been  suggested  by  William  Morgan 
in  the  introduction  to  his  principal  work,  published  in  1779.  A 
much  more  completely  developed  system  had  also  been  devised  by 
the  German  scientist,  Tetens,  and  published  in  1785;  but  his 
system  was  not  known  to  actuaries  until  much  later,  having  been 
buried  in  a  work  upon  another  subject.  In  1812  Francis  Bailly 
presented  to  the  Eoyal  Society  a  paper  by  George  Barrett,  a  self- 
taught  English  mathematician,  in  which  a  system  was  outlined. 
The  paper  was  read,  but  not  published,  by  the  Royal  Society,  an 
omission  which  does  it  little  credit,  which  was  attributed  to  William 
Morgan,  who  was  upon  the  committee,  and  is  supposed  to  have 
been  actuated  by  envy  of  Bailly,  and  in  any  event  did  not  see  the 
importance  of  what  he  had  himself  suggested,  and  which  called 
forth  the  bitterest  criticism  from  Bailly,  who  included  the  paper 
in  the  second  and  subsequent  editions  of  his  own  book.  Unfortu- 
nately, as  a  result  of  the  indifference  of  the  Royal  Society,  Barrett 
was  unable  to  print  the  tables  which  he  had  prepared,  and  his 
invention  for  the  time  was  of  no  practical  use. 


INSURANCE  MATHEMATICS  27 

Griffith  Davies,  himself  a  self-taught  mathematician,  who  had 
risen  by  hard  work  and  his  talents  to  a  responsible  post  as  actu- 
ary of  a  leading  London  company,  developed  the  idea  along  some^ 
what  different  lines  from  those  employed  by  any  of  his  prede- 
cessors, and  by  adding  actual  commutation  tables,  according  to  the 
Carlisle  table  and  various  rates  of  interest,  he  also  made  his  ideas 
of  practical  use  forthwith. 

In  the  same  year  Benjamin  Gompertz,  a  great  mathematician, 
who  otherwise  gave  little  attention  to  actuarial  problems,  wrote 
a  letter  to  Francis  Bailly  concerning  a  mathematical  force  of 
mortality  which,  according  to  his  view,  could  be  given  expression 
in  terms  of  the  infinitesimal  calculus  and  algebra,  and  which  he 
believed  to  be  at  the  foundation  of  all  mortality  tables.  In  his 
letter  he  explained  that  the  natural  law,  which  he  considered  to 
be  expressing  itself  in  the  mathematical  law,  was  that  as  men 
grow  older  the  vitality  wanes,  and  that  the  measure  of  this  change 
caused  the  change  in  the  death  rate  and  could  be  evolved  out 
by  differential  methods.  Thus  within  the  first  twenty-five  years 
of  the  new  century  the  supremacy  of  the  Northampton  table, 
which  at  the  beginning  of  that  century  seemed  so  securely  fixed 
in  that  position,  was  threatened  by  two  foes,  one  the  Carlisle  table, 
which  disputed  its  data,  and  one  Gompertz's  discovery,  which  re- 
newed the  interest  in  De  Moivre's  original  theory  of  a  mathemat- 
ical law  for  mortality,  which  theory  had  been  regarded  as  hope- 
lessly dead.  But  while  the  Carlisle  table  was  making  its  way 
rapidly,  the  theory  of  Gompertz,  being  found  to  square  with  the 
facts  of  none  of  the  tables,  was  for  a  time  thrust  aside. 

In  1826  the  eminent  British  mathematician,  Charles  Babbage, 
published  a  work  entitled  "A  Comparative  View  of  Various  Life 
Offices,"  which  showed  a  marked  bias  for  the  Equitable.  The  book 
is  mainly  non-mathematical,  however. 

In  1832  T.  E.  Edmonds  published  a  book  in  which  he  further 
discussed  the  possibility  of  employing  a  mortality  table  in  which, 
while  it  accurately  represented  experience,  the  graduation  was 
according  to  a  mathematical  formula.  His  book  was  entitled  ^T.ife 
Tables,  Founded  upon  the  Discovery  of  a  Numerical  Law  Eegu- 
lating  the  Existence  of  Every  Human  Being,  Illustrated  by  a  New 
Theory  of  the  Causes  Producing  Health  and  Longevity."     The 


28  DEVELOPMENT  OF 

system  proposed  by  Mr.  Edmonds  bears  a  striking  resemblance 
to  that  suggested  by  the  originator  of  actuarial  science,  John  del 
Witt,  whose  essay  upon  the  subject  was  at  that  time  sleeping  un- 
known to  all  actuaries  in  the  archives  of  Holland  and  West  Fries- 
land.  Edmonds'  system  was  not  adopted,  but  the  ideas  which  are 
set  forth  in  the  book,  and  also  are  plainly  indicated  by  the  singu- 
lar title,  have  survived,  and,  indeed,  found  lodgment  in  many 
minds.  As  Gompertz  had  thought  of  the  thing,  "force  of  mor- 
tality," Edmonds  thought  of  the  name,  and  while  Edmonds'  for- 
mulas are  forgotten,  his  name  for  the  tendency  which  he  did  not 
accurately  measure  has  become  a  permanent  addition  to  life  in- 
surance terminology.  Already,  before  Edmonds,  Thomas  Young 
espoused  the  cause  of  the  mathematical  formula  in  a  letter  which, 
much  to  William  Morgan's  disgust,  was  accepted  and  printed  by  the 
Eoyal  Society  in  1826.  It  was  entitled  "A  Formula  for  Expressing 
the  Decrement  of  Life,"  and  it  led  to  a  controversy  with  the 
nephew  of  the  great  Eichard  Price,  who  considered  that  the  repu- 
tation of  himself  and  his  uncle  depended  upon  refuting  all  argu- 
ments in  favor  of  a  mechanical  system. 

The  Northampton  table  was  giving  way,  however,  before  an 
enemy  more  formidable  than  the  mathematical  formula;  that 
enemy  was  experience.  For  the  British  government  found  its 
experience  with  annuitants  disastrous,  having  sold  annuities  at 
prices  fixed  by  this  table;  and  life  insurance  companies  in  general 
found  the  table  to  absurdly  exaggerate  the  losses  which  they  might 
expect.  Consequently  the  Carlisle  table  steadily  made  its  way  into 
favor.  The  publication  of  the  government  annuity  experience, 
tabulated  by  John  Finlaison  in  1823,  and  adopted  by  the  govern- 
ment in  1829,  completely  displaced  the  Northampton  table  as  to 
its  use  for  annuities,  and,  indeed,  it  was  thereafter  retained 
by  no  company  of  importance  excepting  the  Equitable,  which  stub- 
bornly retains  it,  even  to  this  day,  or  at  least  was  doing  so  at  a 
recent  date. 

Another  consideration  which  occurred  to  de  Witt  at  the  very 
beginning  forced  itself  upon  men's  notice  in  connection  with  an- 
nuitants, and  that  was  that  there  is  a  very  great  measure  of  self- 
selection  among  annuitants,  assuring  that  the  average  of  longevity 
will  be  much  higher  than  among  the  population,  and  also  than 


INSURANCE  MATHEMATICS  29 

among  insured  lives,  in  spite  of  the  careful  medical  selection  which 
companies  had  learned  to  make.  This  made  all  tables  that  are 
suitable  for  insured  lives  unsuitable,  for  that  very  reason,  to 
measure  the  value  of  annuities.  The  fact  was  recognized  in  France 
before  it  was  in  Great  Britain,  for  in  France  the  Duvillard  table, 
named  after  its  author,  was  introduced  so  early  as  180G,  and  being 
better  suited  to  the  purpose  than  DesParcieux's  table,  was  adopted 
for  annuity  purposes.  In  Great  Britain  the  government  experi- 
ence table  of  1829  was  employed  until  1860,  when  a  second  gov- 
ernment table  appeared,  and  this  in  succession  until  1883,  when  a 
third  government  table  appeared,  all  the  production  of  the  Fin- 
laisons,  father  and  son.  At  this  time  an  unadjusted  table  has  just 
appeared,  drawn  from  the  joint  experience  with  annuitants  of  the 
chief  British  companies;  and  while  it  confirms  in  a  remarkable 
manner  the  lessons  of  the  government  tables,  it  shows  even  a 
greater  longevity  and  will  doubtless  be  mainly  employed  here- 
after. 

In  Edmonds'  work,  in  addition  to  discussing  the  force  of  mor- 
tality, he  considered  the  force  of  sickness,  i.  e.,  the  liability  to 
disablement,  which  he  deemed  to  be  subject  to  the  same  law,  be- 
ing only  another  expression  of  it.  There  appeared  in  1835  a  work 
entitled  "A  Treatise  on  Friendly  Societies,'*'  by  Charles  Ansell, 
which  contained  the  experience  of  the  Highland  Society  in  rela- 
tion to  sickness,  and  also  the  combined  experience  of  a  number 
of  Scottish  friendly  societies.  Ansell  supplied  monetary  tables 
for  the  same.  These  were  adopted  as  standard  by  the  friendly 
societies  generally,  and  as  they  proved  to  be  too  low,  much  em- 
barrassment was  caused  thereby.  Xo  trace  of  evidence  that  Ed- 
monds' theory  was  correct  was  discernible,  however,  in  this;  nor, 
indeed,  could  its  correctness  be  tested  before  there  were  several 
experiences  at  hand  to  compare.  The  publication  of  Henry  Rat- 
cliff's  report  upon  the  experience  of  the  Manchester  Unity  of  Odd 
Fellows  for  1846-8,  and  of  the  report  upon  the  experience  of  the 
Foresters,  by  Frederick  G.  P.  Neison,  at  about  the  same  time, 
opened  the  way  for  a  thorough  development  of  the  subject.  Two 
later  reports  of  the  Manchester  Unity  appeared  during  Mr.  Rat- 
cliff's  lifetime,  viz.,  one  covering  1856-60,  and  the  other  1866-70; 
and,  as  a  result  of  careful  comparison,  it  was  developed  that,  so 


30  DEVELOPMENT  OF 

far  as  could  be  determined  a  priori,  the  liability  to  illness  could 
be  represented  as  a  function  of  the  force  of  mortality.  The  very 
recent  investigation  of  the  entire  aggregate  experience  of  the 
British  friendly  societies,  during  the  present  year,  confirms  also 
the  theory  that  from  a  mortality  table  the  liability  to  sickness  may 
be  ascertained  with  much  accuracy.  It  is  still,  however,  a  matter  of 
ratio,  and  the  algebraic  expression  for  a  force  of  sickness  has  not  as 
yet  been  developed.  Yet  another  report  upon  later  experience  of 
the  Manchester  Unity  is  now  in  preparation. 

With  the  advent  of  the  commutation  method  of  making  com- 
putations there  was  naturally  a  great  change  in  symbols,  and  a 
new  notation  began  to  come  in.  In  consequence  of  this,  and  of 
the  new  formulas  as  well,  that  were  found  necessary,  the  old  text- 
books were  rapidly  becoming  antiquated.  Naturally,  therefore. 
Prof,  de  Morgan's  "An  Essay  on  Probabilities  and  on  Their  Appli- 
cation to  Life  Contingencies  and  Insurance  Offices,"  published  in 
1838,  and  especially  "A  Treatise  on  the  Value  of  Annuities  and 
Eeversionary  Payments,"  by  David  Jones,  published  in  1840,  were 
generally  welcomed.  The  former  remained  for  a  long  time  the 
best  English  text-book  on  probabilities,  though  not  so  highly  es- 
teemed as  to  its  actuarial  content,  while  the  latter  was  for  more 
than  a  quarter  century  the  book  of  books  for  the  actuarial  stu- 
dent, and  is  valued  unto  this  day. 

The  years  from  1840  to  1860  were  fruitful  in  the  elucidation 
of  many  questions  which  were  very  difficult  to  handle  before  the 
commutation  system  was  introduced.  They  also  saw  the  begin- 
ning of  the  great  work  of  England's  registrar-general.  Dr.  Will- 
iam Farr,  author  of  English  Life  Tables  Nos.  1,  2  and  3,  the  first 
two  of  which  appeared  during  this  period.  Toward  the  close  of 
the  period  the  Institute  of  Actuaries  was  organized  in  London, 
and  although  this  body  was  frowned  upon  by  many  of  the  actua- 
ries of  great  reputation  in  that  day,  who  refused  to  join,  and  or- 
ganized the  Actuaries'  Club  as  a  more  exclusive  and  aristocratic 
society,  the  Institute,  by  setting  its  standard  higher  and  higher, 
and  also  by  excluding  all  considerations  as  to  candidates  except 
those  of  fitness,  won  a  pre-eminent  place  for  itself  and  also  great 
respect  for  the  entire  profession.  It  also  founded  a  journal  in 
which  from  that  time  forward  the  most  valuable  discoveries  and 


INSURANCE  MATHEMATICS  31 

inventions  have  usually  been  first  given  to  the  world.  The  his- 
tory of  the  progress  of  actuarial  science  may  be  read  since  its 
foundation  in  the  volumes  of  this  journal. 

To  this  "Journal  of  the  Institute  of  Actuaries"  William  Make- 
ham,  in  1860,  contributed  what  has  proved  to  be  a  solution  of  the 
disagreement  between  the  school  that  followed  De  Moivre  and 
the  school  that  followed  Dr.  Price.  In  other  words,  he  discovered 
and  gave  to  the  profession  the  extension  of  the  principle  laid 
down  by  Gompertz,  which  was  needed  to  reconcile  his  theory  as 
to  the  force  of  mortality  with  the  facts  of  experience.  Gompertz 
had  guessed  that  increasing  mortality  was  the  consequence  of  wan- 
ing vitality,  and  could  be  measured  by  a  steadily  increasing  func- 
tion. Makeham  divined  that  deaths  might  be  divided  into  two 
classes,  viz.,  from  causes  that  apply  with  almost  equal  force  at 
all  ages,  such  as  accidents,  and  from  causes  which  flow  from  di- 
minished vitality.  A  modification  of  the  formula  of  Gompertz 
to  agree  with  this  idea  gives  an  expression  for  the  force  of  mor- 
tality which  has  proved  most  useful,  and  also  to  measure  with  great 
accuracy  the  facts  of  experience. 

Notwithstanding  this  discovery,  so  strong  was  the  force  of  tra- 
dition and  the  influence  of  the  school  of  Dr.  Price  that  when  a 
table  of  mortality  was  constructed  from  the  experience  of  20 
British  companies,  under  the  auspices  of  the  Institute,  the  work 
being  completed  and  published  in  1869,  the  same  was  graduated 
by  a  formula  which  was  really  merely  a  very  smooth  mathematical 
expression  for  the  graphic  method;  and  it  was  not  before  1887 
when  the  second  volume  of  the  "Institute  of  Actuaries'  Text-book*' 
was  published,  that  tables  were  published  in  Great  Britain  which 
were  graduated  by  Makeham's  formula.  It  was  found  that  such 
graduation  gave  results  very  close  to  the  original  data,  and  the 
system  is  now  practically  universally  accepted.  It  presents  all 
the  advantages  in  the  matter  of  facility  of  computing  joint  life 
functions  that  were  originally  foreseen  by  Simpson  and  others. 

The  20  British  Offices  table  was  not  the  first  combined  expe- 
rience table  to  be  constructed  in  Great  Britain.  In  1843  a  table, 
known  in  Great  Britain  as  the  17  Offices,  and  in  America  as  the 
Actuaries',  or  Combined  Experience  Table,  was  published,  having 
been  imder  construction  for  a  number  of  years.     It  was  hotly 


32  DEVELOPMENT  OF 

attacked  in  England  by  both  the  friends  of  the  Carlisle  table  and 
the  lingering  champions  of  the  Northampton  table,  and  it  never 
became  standard  there;  though  tables  of  monetary  values  were 
published  by  Jenkin  Jones. 

In  the  United  States,  on  the  contrary,  this  table,  chancing  to 
have  come  out  at  about  the  right  moment,  and  appealing  to  Elizur 
Wright  as  a  fair  test  of  solvency,  was  introduced  by  him  in  1859 
as  a  standard  for  valuing  the  policies  of  companies  doing  busi- 
ness in  Massachusetts,  of  which  State  he  was  commissioner,  in 
consequence  this  table  was  pretty  widely  adopted  by  American 
companies  and  departments,  and  is  yet  in  very  common  use. 

Early  in  the  sixties  there  was.  published  a  table  drawn  from  the 
experience  of  the  Mutual  Life  Insurance  Company.  This  table 
was  graduated  by  Sheppard  Homans,  actuary  of  that  company. 
A  few  years  later  Mr.  Homans  published  another  table  embracing 
a  modification  of  the  Actuaries'  table,  by  the  experience  of  the 
Mutual  Life,  and  with  some  other  changes.  This  table  was  given 
the  name  American  Experience  Table,  and  was  adopted  as  the 
standard  in  New  York  and  in  other  States,  as  well  as  by  the  Mutual 
Life  and  many  companies.    It  is  now  also  in  common  use. 

As  in  Great  Britain,  so  also  in  the  United  States,  the  first 
table  giving  the  combined  experience  of  the  home  companies  has 
been  discredited  and  little  used.  In  1870  a  table  covering  the 
experience  of  36  American  offices,  which  had  been  prepared  under 
the  direction  of  the  Chamber  of  Life  Assurance  and  a  committee 
of  prominent  actuaries,  made  its  appearance.  Although  it  was 
graduated  according  to  Makeham's  formula,  and  thus  afforded 
many  advantages  in  actuarial  operations,  and  although  it  was 
accompanied  also  by  a  most  elaborate  set  of  monetary  tables,  it 
was  adopted  by  no  company  of  consequence,  and  has  never  been 
standard  anywhere. 

The  principal  contributions  of  America  to  actuarial  mathe- 
matics have  been  three  formulas  for  the  computation  of  individual 
reserves  by  what  is  known  as  the  retrospective  method,  which  are 
known  by  the  names  of  their  authors,  as  follows :  Wright's,  Fack- 
ler's  and  McClintock's  formulas.  The  first  two  are  more  com- 
monly employed  to  compute  values  of  policies  involving  one  life 
only,  and  the  last  to  compute  the  values  of  joint  life  policies, 


INSURANCE  MATHEMATICS  33 

for  which  purpose  it  is  singularly  useful.  Another  important 
discovery  was  the  contribution  plan  of  dividing  surplus,  the  honor 
of  which  is  due  to  Sheppard  Homans  and  David  Parks  Fackler, 
jointly. 

The  most  recent  developments  in  the  mathematics  of  insur- 
ance are  due  to  one  man,  who  has  by  competent  authority  been 
pronounced  the  greatest  living  actuary.  Dr.  Thomas  Bond  Sprague. 
They  are  the  discovery  that  a  select  mortality  table,  instead  of 
the  mortality  tables  commonly  in  use,  represents  most  closely  the 
iacts  of  experience  in  life  insurance  companies.  Thus  the  usual 
form  of  mortality  table  exhibits  the  average  death  rate  at  a  cer- 
tain age,  for  instance,  without  regard  to  whether  the  lives  are 
freshly  selected  by  medical  examination  or  not.  It  has  long  been 
understood  that  the  mortality  during  early  years  of  insurance  is 
comparatively  light,  and  the  theory  of  adverse  selection  by  dis- 
continuances had  been  exploited  long  before  Dr.  Sprague's  time. 
Indeed,  in  the  analysis  which  actuaries  made  of  the  mortuary  ex- 
perience of  the  Equitable  Society  when  first  published,  they  clear- 
ly distinguished  the  effects  of  adverse  selection  for  twenty  years 
after  entry.  But  Dr.  Sprague  discovered  that,  in  the  later  expe- 
rience of  companies,  embodied  in  the  data  for  the  20  Offices'  table 
the  effects  of  favorable  selection  appeared  to  wear  off  in  about 
five  years.  So  he  recommended  the  use  of  select  tables,  com- 
posed of  the  actual  experience  during  the  first  five  years  of  in- 
surance, combined  with  the  general  experience  upon  lives,  admitted 
more  than  five  years.  He  prepared  and  published  the  monetary 
tables  necessary  to  give  his  views  effect. 

This  was  followed  some  years  later  by  another  discovery  and 
suggestion  of  even  greater  practical  importance,  viz.,  that  the 
net  reserve  system,  as  usually  employed,  was  inaccurate,  and  should 
be  modified.  Thus  he  discovered  that,  owing  to  the  circumstance 
that  expenses  are  much  higher  the  first  year  of  insurance  than 
later,  the  reserve  which  the  usual  methods  brought  out  was  neither 
just  according  to  the  fundamental  meaning  of  the  prospective 
nor  of  the  retrospective  formulas,  because  it  would  not  be  needed 
to  make  good  the  deficiencies  of  future  premiums,  the  original 
premiums  being  adequate  for  all  purposes;  nor  could  it  have  been 
accumulated  from  past  premiums,  the  conditions  being  as  per 


34  DEVELOPMENT  OF  INSURANCE  MATHEMATICS 

assumption,  and  expenses  being  incident,  as  they  actually  arc. 
He  presented  his  views  before  British  actuaries  at  various  times, 
and  at  length  before  the  first  Congress  of  Actuaries  in  1895.  The 
issues  which  he  raised  are  now  being  fought  out  by  American 
companies  and  departments,  mainly  in  the  courts,  in  disputes  as 
to  construction  of  contracts,  but  also  between  opposing  schools  of 
actuaries.  In  France  and  some  other  countries  the  principles 
laid  down  by  Dr.  Sprague  have  already  been  adopted. 

In  the  foregoing,  upon  looking  it  over,  I  find  that  I  have 
forgotten  to  mention  the  invention  of  a  system  of  conversion  of 
annuities  to  insurknce,  and  vice  versa,  and  the  production  of  a 
most  remarkable  series  of  conversion  tables  for  annuities  and  life 
insurances,  which  was  printed  in  1856.  It  was  the  work  of  Will- 
iam Orchard,  one  of  the  most  brilliant  young  mathematicians  of 
Great  Britain,  whose  early  death  was  justly  deplored.  I  have  also 
forgotten  to  mention  the  work  of  David  Chisholm,  whose  series 
of  tables  have  been  most  valuable  aids  to  actuaries. 

There  has  been  no  lack  of  text-books  in  the  last  half  of  the 
nineteenth  century.  The  Institute  of  Actuaries  followed  up  its 
great  work  of  collecting  the  combined  mortality  experience  of 
twenty  companies  into  a  mortality  table  by  putting  forth  a  text- 
book which  is  now  everywhere  the  standard  authority  for  higher 
education  upon  the  subject.  It  was  published  in  two  volumes,  the 
first  dealing  with  annuities  certain  and  the  second  dealing  with 
life  contingencies.  A  second  edition  of  the  first  volume  is  now 
about  to  appear,  and  a  second  edition  of  the  second  volume  is 
m  preparation.  In  America  there  have  also  been  published  text- 
books, though  none  which  even  pretends  to  the  thoroughness  of 
the  Institute  of  Actuarh 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


^^^(H  .2:^ 


.1*1. 


JAU  8^959  88 


^ 


^ 


i0 


za^ 


W^t,?ffloc. 


JUL3119^ 


LD  21-100m-8,'34 


.•1*-  ^^' 


/    i\ 


